The thesis is constituted in two parts. The first part including the first four chapters concentrates on the surface mapping class groups. The second part including the last three chapters focuses on their applications in the 3-manifold theory. Chapters I, II, and III show a new set of three generators, L, N and T of the surface mapping class groups M\sbg and investigate their topological and algebraic properties. Chapter IV finds a finite set of generators of the subgroup K\sbg, which consists of the mapping classes that can be extended to a solid handlebody, in words of the generators of M\sbg given in the earlier chapters. Chapter V describes all Heegaard decompositions of the 3-sphere, and relates the homology 3-spheres to the elements of the Torelli subgroups. Chapter VI presents a new proof of the fundamental theorem of Kirby calculus on links by using the presentation of M\sbg. The most important result of the thesis, which answers a question asked long ago about the stable equivalence of Heegaard decompositions of 3-manifolds, is proved in Chapter VII. We quote it here: Theorem VII.1.1. Any two Heegaard decompositions of the same genus of a 3-manifold of genus g are stably equivalent by adding no more than 3g - 3 trivial handles.